Mathematics 535: Introduction to Algebraic Geometry, Fall 2008

Movie of the week - Week 3

The images above comprise the frames of the Week 3 algebraic geometry
movie of the week. Each is an affine part of the real locus of the
projection of the Veronese surface from projective 5 space to
projective 3 space. The Veronese surface is the image of the
projective plane using the 6 degree 2 monomials. The image of
projection from a line in P^5 gives a projective
variety in P^3. The real locus of the intersection of this projection
with an affine 3 space is drawn in the pictures above, which are all
projectively inequivalent over the complex field. Any complex projection
is projectively equivalent to one of the 6 above
or lies in a plane in 3 space. The green lines are contained
in the surface (any line hitting a hypersurface in P^3 more times than
its degree is wholly contained in the surface), and are the singular
locus of the variety, that is points where the gradient of the
defining polynomial vanishes.

These are all related to examples of Steiner surfaces, which are the images of
the real projective plane in P^3 given by 4 independent homogeneous
polynomials of degree 2 in 3 variables. The real locus of the
projection contains the projection of the real locus, but in some
cases is strictly bigger. In these pictures any green lines not
in the main surface are in fact not in the projection of the real
projective plane, but are in the real locus of the projection of the
Veronese surface. The left most is Steiner's
Roman surface, while the second from right is the Caley ruled cubic surface.
Visit A. Coffman's
Steiner Surface page for a nice source of information on these
objects, as well as a discussion of the classification over the real
field.

The general projection will yield a variety projectively equivalent to
the Roman surface. When projecting from a line L in P^5 which misses
the Veronese surface to a dimension 3 linear space disjoint from the
line we compute where the plane determined by L and a point on
the Veronese hits the dimension 3 space. This map is one to one
except when a plane hits the Veronese in more than one point, in
which case the secant line joining these points must hit L at a point
where L hits the set of all secant
lines. This secant variety is a hypersurface in P^5 given by a degree 3 homogeneous
polynomial (see Harris, page 145), so there are 3 points on L that
also lie on a secant to the Veronese surface. For each of these three
points there is a P^1 of lines through them that hit the Veronese
2 times, and they all live in a plane hitting L only at 1 point.
(see Harris picture on page 144). For planes determined by L
and these lines we get 2 points on the Veronese mapping to the same
point on the projection, and these image points lie in a linear P^1
in three space. Thus the general projection to P^3 will have 3
double lines, which by similar logic will intersect in a point. This
explains geometrically the 3 green lines meeting at the center of the
picture of Steiner's Roman Surface.

We list the 6 different lines that we project from to get the 6
examples above. See the Macaulay worksheet 3 to see how to compute
the equations of the projections.
Take the Veronese as the set of points
[z_0,...,z_5]=[u_0^2,u_1^2,u_2^2,u_0u_1,u_0u_2,u_1u_2].
The lines are

z_0+z_1+z_2=z_3=z_4=z_5=0

z_3=z_4=z_2-z-1=z-0-z_1+z_2=0

z_0+2z_1+z_2=2z_1+z_2=z_2+z_4=z_3+z_5=0

z_3=z_4=z_5=z_1-z_2=0

z_3=z_4-z_1=z_5=z_2=0

z_0+z_1+z_2=z_3=z_4=z_0-z_1-z_2=0

The last is described topologically by identifying the equator to a
point in the upper hemisphere model of the projective line, which
yields a topological sphere.